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To estimate the derivative \( f'(2.1) \) for the function \( f(x) = 1.5^{1.6x} \), we can use the definition of the derivative with a small interval. This approach is known as the difference quotient:

\[
f'(2.1) \approx \frac{f(2.1 + h) - f(2.1)}{h}
\]

We need to choose a small value for \( h \) (typically \( h = 0.01 \) or smaller) to ensure an accurate estimate. Let's calculate it using \( h = 0.01 \) and the function \( f(x) = 1.5^{1.6x} \).

First, we calculate \( f(2.11) \) and \( f(2.1) \):

\[
f(2.11) = 1.5^{1.6 \times 2.11}, \quad f(2.1) = 1.5^{1.6 \times 2.1}
\]

Then apply the difference quotient formula:

\[
f'(2.1) \approx \frac{f(2.11) - f(2.1)}{0.01}
\]

I'll compute this next.The estimated derivative \( f'(2.1) \) using the difference quotient method with \( h = 0.01 \) is approximately \( 2.54 \).

Would you like a more detailed explanation of the steps, or do you have any further questions?

Here are five related questions to expand your understanding:
1. How does the choice of \( h \) affect the accuracy of the derivative estimate?
2. What is the exact derivative of the function \( f(x) = 1.5^{1.6x} \)?
3. How would you compute the derivative at a different point, like \( f'(1.5) \)?
4. What is the geometric interpretation of the derivative at \( x = 2.1 \)?
5. How can you improve the precision of this estimate using smaller intervals?

**Tip:** Using a smaller \( h \) in the difference quotient will generally increase the accuracy of your estimate. However, if \( h \) becomes too small, numerical errors may arise.

Let f(x) = 1.5^(1.6x). Estimate the derivative f'(2.1) to within two decimal places by using a small enough interval.

Learn how to estimate the derivative f'(2.1) for the exponential function f(x) = 1.5^(1.6x) using the difference quotient method. This example uses h = 0.01 and provides a detailed calculation of the derivative to two decimal places.

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